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Chapter 6: Problem 91

Find the function value. Round to four decimal places. $$\cos \left(-295.8^{\circ}\right)$$

Short Answer

Expert verified
0.4346

Step by step solution

01

Understand the Problem

The goal is to find the value of \(\text{cos}(-295.8^\text{掳})\) and round it to four decimal places.
02

Convert the Negative Angle to a Positive Angle

Angles can be converted by adding 360掳 until the angle is positive. \(\text{cos}(-295.8^\text{掳} + 360^\text{掳}) = \text{cos}(64.2^\text{掳})\). This makes the calculation easier.
03

Compute the Cosine Value

Use a calculator to find \(\text{cos}(64.2^\text{掳})\). Ensure the calculator is set to degrees, not radians.
04

Round the Value

After computing \(\text{cos}(64.2^\text{掳})\), round the result to four decimal places. The value should be approximately 0.4346.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosine Function
The cosine function, often denoted as \(\text{cos}\), is one of the primary trigonometric functions used in various fields like mathematics, physics, and engineering. It relates the angle of a right triangle to the ratio of the length of the adjacent side over the hypotenuse.
The function can be written as:
\[\text{cos}(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}\].
Cosine values range from -1 to 1. These values repeat periodically as the angle increases or decreases. This periodicity is a feature of all trigonometric functions.
Angle Conversion
Understanding angle conversion is crucial when dealing with trigonometric functions. Angles can be positive or negative, measured in degrees or radians. A standard technique for converting angles is to add or subtract 360掳 (full circle) until you get a positive measure:
  • In our example, we have \(-295.8^\text{掳}\).
  • Add 360掳 to -295.8掳 to convert it into a positive angle: \(-295.8^\text{掳} + 360^\text{掳} = 64.2^\text{掳}\).
Hence, the problem reduces to finding \(\text{cos}(64.2^\text{掳})\).
Converting angles can simplify the problem, as positive angles within one full circle (0掳 to 360掳) are easier to handle.
Significant Figures
Significant figures are essential when dealing with trigonometric calculations, especially in precision-required tasks like engineering and physics.
When rounding numbers to a specified number of significant figures, every digit counts except:
1. Leading zeros
2. Trailing zeros in whole numbers without a decimal
For our example:
  • First, calculate \(\text{cos}(64.2^\text{掳})\) using a calculator set to degrees.
  • You will get a value close to 0.43459.
  • Round this value to four decimal places, resulting in 0.4346.
Always ensure that rounding does not distort the intended precision. This is vital for getting accurate results in problems.

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Most popular questions from this chapter

Given the function value and the quadrant restriction, find \(\theta\). FUNCTION VALUE = \(\cos \theta=-0.0990\) INTERVAL = \(\left(90^{\circ}, 180^{\circ}\right)\) \(\boldsymbol{\theta}\) = ____

Use a graphing calculator to graph the function. $$y=\cos x-x$$

Find the reference angle and the exact function value if they exist. $$\cos 0^{\circ}$$

Determine the domain of the function. $$f(x)=\frac{\sin x}{\cos x}$$

Satellite Location.\(\quad\) A satellite circles the earth in such a way that it is \(y\) miles from the equator (north or south, height not considered) \(t\) minutes after its launch, where $$ y(t)=3000\left[\cos \frac{\pi}{45}(t-10)\right] $$ (IMAGE CAN'T COPY)
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